Implicitly Defined Functions
Tom Green
7 min read
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Study Guide Overview
This study guide covers implicitly defined functions, contrasting them with explicit functions. It explains how to graph and solve for variables in implicit equations, emphasizing domain and range. The guide also discusses interpreting slope (positive, negative, zero, undefined) and identifying horizontal/vertical intervals. Finally, it provides practice questions covering these concepts.
#Implicitly Defined Functions: Your Night-Before-the-Exam Guide
Hey there, future AP Pre-Calculus master! Let's dive into implicitly defined functions. Think of this as your cheat sheet for tonight – clear, concise, and ready to boost your confidence. Let's get started!
#What are Implicitly Defined Functions?
An equation with both 'x' and 'y' can secretly describe one or more functions. The graph of such an equation is simply all the (x, y) pairs that make the equation true. 🧑🎨
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Graphing Equations with x and y
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Explicit Functions: You're used to these! Like
y = 2x + 1. Plug in 'x', get 'y'. Graph is a line. ↗️
- Caption: The graph of the function y = 2x + 1, a simple linear equation.
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Implicit Functions: Equations where 'x' and 'y' are mixed, like
x² + y² = 1. You can't easily isolate 'y'. 💡
- Caption: The graph of the function x² + y² = 1, a circle centered at the origin.
Think of it like this: Explicit functions are like having a recipe where you directly get the output (y) from the input (x). Implicit functions are like a puzzle where you have to figure out the relationship between x and y to find the solutions. 🧩
#Solving for One Variable
- Solving for 'y' can give you one or more functions, each representing a piece of the original graph. For
x² + y² = 1, solving for 'y' givesy = ±√(1 - x²), two functions (top and bottom halves of the circle). 〽️ - S...
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